Problem 42 - Precalculus with Limits

Question

In Exercises 19-42, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. \(f(x) = |x-1|\)

Step-by-Step Solution

Answer

The graph of the function \(f(x) = |x-1|\) is a V-shaped graph intersecting the x-axis at x=1. It is symmetrical around the line x=1 and does not go below the x-axis.

Step 1: Understand the Absolute Function

Absolute function, \(f(x) = |x-1|\) is defined as \(f(x) = x - 1, if x≥1\) and \(f(x) = 1 - x, if x < 1\). This means, the graph of this function will intersect the x-axis at point x=1 and will be a mirror image around x=1.

Step 2: Choose the Window of the Graph

Using a graphing utility, select a value for x that ranges from -10 to 10. Similarly, for y, a range from -10 to 10 should be fine as well. This is because absolute values are always positive and we do not expect the function to return any y-values less than zero. Also, since the graph is a mirror image around x=1, ranging from -10 to 10 for x will show the symmetrical behavior of the graph clearly.

Step 3: Graph the Function

Enter the function \(f(x) = |x-1|\) into the graphing utility and plot the graph with the chosen window parameters. You will see the V-shaped graph that intersects the x-axis at x=1. This graph illustrates the typical nature of all absolute value functions.

Key Concepts

Absolute Value Functions

Absolute value functions are interesting and versatile mathematical expressions. The basic notion of an absolute value is that it describes a number's distance from zero on the number line, regardless of direction. Simply put, an absolute value function features variables within absolute value brackets, such as \( f(x) = |x - 1| \).
These types of functions typically exhibit a V-shaped graph. Here's how this works:

• The expression inside the absolute value can result in both positive and negative numbers.
• However, any negative result is transformed into its positive equivalent when the absolute value is applied.
• The turning point or "V" vertex of the function \( f(x) = |x - 1| \) intersects the x-axis at \( x = 1 \), because at this point, the expression inside the absolute value is zero.

The vertex and symmetrical nature make these functions distinct and easily recognizable when graphed. Understanding these principles helps predict the shape and behavior of the absolute value functions when using graphing utilities.

Graphing Utilities

Graphing utilities are powerful tools used to visualize functions. They are software or devices that help us see how functions look on a coordinate plane. Using a graphing utility can significantly aid in understanding a function's characteristics.
For an absolute value function, input the expression into the graphing utility, such as a calculator or a software program. When entering the function \( f(x) = |x - 1| \):

• Carefully include the absolute value notation, which is critical for graph accuracy.
• Ensure no syntax errors to prevent incorrect graphs.
• The utility will then plot the characteristic V-shape graph, visualizing how the function behaves across different values of \( x \).

By observing the graph, you can see real-time changes and understand the relationship between \( x \) and \( y \) values. This process fosters a deeper comprehension of mathematical functions beyond what static images offer.

Viewing Window Selection

Setting an appropriate viewing window is crucial when graphing functions using utilities. The viewing window dictates the range of x and y values visible on the graph. Selecting a range that encompasses the key features of the function ensures a complete view.For the function \( f(x) = |x - 1| \), you want to include the V shape and the vertex:

• A good starting point is to have a range for both \( x \) and \( y \) from -10 to 10. This selection captures the essential elements of the function while maintaining symmetry around \( x = 1 \).
• The window should be wide enough to see the graph's complete V-shape but not too broad that it obscures the details.
• Adjusting the window helps focus on particular features, like close-ups of the vertex or extended views of the tails.

Experiment with different settings if necessary to balance broad coverage and the specific details needed for analysis. Proper window adjustments enhance your ability to study the graph's behavior effectively.